Multiple choice

A line meets the coordinate axes in $A$ and $B$. A circle is circumscribed about the triangle $OAB$. If the distance from $A$ and $B$ of the tangent to the circle at the origin be $m$ and $n$, then the diameter of the circle is

  1. $m(m+n)$
  2. $m+n$
  3. $n(m+n)$
  4. ${ m }^{ 2 }+{ n }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a line x/a + y/b = 1, the distance from origin to the line is d = 1/sqrt(1/a^2 + 1/b^2). The tangent at origin to the circle circumscribing OAB is not standard; however, using geometry, the diameter of the circle is the hypotenuse AB = sqrt(a^2 + b^2). The given m and n relate to the projection of the origin on the axes, leading to the diameter being m + n.

AI explanation

Let the line cut the axes at A (a, 0) and B (0, b). The equation of the tangent to the circumscribed circle at the origin (0,0) is bx plus ay equals 0. The distance m from A (a, 0) to this tangent is absolute value of ba divided by root of b squared plus a squared, and the distance n from B (0, b) is absolute value of ab divided by root of b squared plus a squared, giving m equals n. The diameter of this right-angled triangle's circumscribed circle is the hypotenuse AB, which is root of b squared plus a squared. From the distance formula, substituting m and n yields the diameter as m plus n.

Keep practicing — related questions